A-level · Integration

Areas under parametric curvesExam-style questions

Finding the area under a curve given by parametric equations. Why the area is the integral of y times dx/dt with respect to t; turning the x-limits into t-limits; the region a curve encloses with the x-axis; which way round the limits go when x decreases as t increases; curves with trig parameters; and regions with a straight edge, made of a curved piece and a triangle.

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24 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of part of the curve defined by the parametric equations

$x = t^2 - 1 \qquad y = \dfrac{5}{(t + 1)(t + 4)} \qquad t \geqslant 0$

The region $R$ , shown shaded in Figure 1, is bounded by the curve, the $y$ -axis, the $x$ -axis and the line with equation $x = 8$

The curve meets the $y$ -axis where $t = a$ and meets the line with equation $x = 8$ where $t = b$ .
(a) Find the value of $a$ and the value of $b$ .

(2 marks)
(b) Show that the area of region $R$ is given by

$M\int_a^b \dfrac{t}{(t + 1)(t + 4)}\,dt$

where $M$ is a constant to be found.

(5 marks)
(c) (i) Write $\dfrac{t}{(t + 1)(t + 4)}$ in partial fractions.

(ii) Use algebraic integration to find the exact area of $R$ , giving your answer in the form $p\ln\dfrac{7}{5} + q\ln 2$ where $p$ and $q$ are rational constants.

(6 marks)
Level 5 · 13 marksTry it
2
The diagram shows the curve $C$ with parametric equations $x = t^2,\ y = 4t$ , $t \ge 0$ . The point $P$ is where $t = 1$ , and the normal to $C$ at $P$ meets the $x$ -axis at $Q$ . The region $R$ , shown shaded, is bounded by the arc $OP$ (where $O$ is the origin), the line $PQ$ and the $x$ -axis. Find the exact area of $R$ .
Level 5 · 6 marksTry it
3
The curve $C$ has parametric equations $x = t^2,\ y = 6t$ , $t \ge 0$ . The point $P$ is where $t = 2$ , and the normal to $C$ at $P$ meets the $x$ -axis at $Q$ . Find the exact area of the region bounded by the arc $OP$ , the line $PQ$ and the $x$ -axis, where $O$ is the origin.
Level 5 · 6 marksTry it
4
The curve $C$ has parametric equations

$x = 2\cos t, \qquad y = 3\sin t, \qquad 0 \leqslant t \leqslant \frac{\pi}{2}$

Find the exact area of the region bounded by $C$ and the positive $x$ - and $y$ -axes. (6 marks)
Level 5 · 6 marksTry it
5
A curve has parametric equations $x = \mathrm{e}^{2t}$ , $y = t$ . Find the exact area of the region bounded by the curve, the $x$ -axis, and the lines corresponding to $t = 0$ and $t = 1$ . (6 marks)
Level 5 · 6 marksTry it

7 more questions on areas under parametric curves

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